Product rule for differentiation

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Statement for two functions

Verbal statement

If two (possibly equal) functions are differentiable at a given real number, then their pointwise product is also differentiable at that number and the derivative of the product is the sum of two terms: the derivative of the first function times the second function and the first function times the derivative of the second function.

Statement with symbols

Suppose f and g are functions, both of which are differentiable at a real number x = x_0. Then, the product function f \cdot g, defined as x \mapsto f(x)g(x) is also differentiable at x, and the derivative at x_0 is given as follows:

\! \frac{d}{dx} [f(x)g(x)]|_{x = x_0} = f'(x_0)g(x_0) + f(x_0)g'(x_0)

or equivalently:

\! \frac{d}{dx} [f(x)g(x)]|_{x = x_0} = \frac{d(f(x))}{dx}|_{x=x_0} \cdot g(x_0) + f(x_0)\cdot \frac{d(g(x))}{dx}|_{x = x_0}

If we consider the general expressions rather than evaluation at a particular point x_0, we can rewrite the above as:

\! \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

or equivalently:

(f \cdot g)' = (f' \cdot g) + (f \cdot g')